On the Dimension of Secant Varieties
Luca Chiantini, Ciro Ciliberto
TL;DR
This paper extends Zak's foundational theorems to establish sharp lower bounds on the dimensions of higher secant varieties of algebraic varieties, providing classifications for cases where bounds are tight.
Contribution
It generalizes Zak's theorems on tangencies and linear normality, and classifies varieties attaining the lower bounds for secant variety dimensions.
Findings
Established sharp lower bounds for secant variety dimensions
Classified varieties where bounds are attained
Extended Zak's theorems to broader contexts
Abstract
We generalize Zak's theorems on tangencies and on linear normality as well as Zak's definition and classification of Severi varieties. In particular we find sharp lower bounds for the dimension of higher secant varieties of a given variety X under suitable regularity assumption on X, and we classify varieties for which the bound is attained.
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Taxonomy
TopicsTensor decomposition and applications · Meromorphic and Entire Functions · Algebraic Geometry and Number Theory